\( \left. \begin{array} { l }{ ( a + b ) ( a - b ) ( a ^ { 4 } + a ^ { 2 } b ^ { 2 } + b ^ { 4 } ) } \\ { = ( a ^ { 2 } - b ^ { 2 } ) ( a ^ { 4 } + a ^ { 2 } b ^ { 2 } + b ^ { 4 } ) } \\ { = a ^ { 6 } - b ^ { 8 } } \end{array} \right. \)

Let's analyze the algebraic expression you've provided step by step to ensure its correctness. **Given Expression:** \[ \left. \begin{array} { l } { ( a + b ) ( a - b ) ( a ^ { 4 } + a ^ { 2 } b ^ { 2 } + b ^ { 4 } ) } \\ { = ( a ^ { 2 } - b ^ { 2 } ) ( a ^ { 4 } + a ^ { 2 } b ^ { 2 } + b ^ { 4 } ) } \\ { = a ^ { 6 } - b ^ { 8 } } \end{array} \right. \] Let's break this down: 1. **First Step: Expansion of \((a + b)(a - b)\)** \[ (a + b)(a - b) = a^2 - b^2 \] This step is correct and uses the difference of squares formula. 2. **Second Step: Multiplying \((a^2 - b^2)\) by \((a^4 + a^2b^2 + b^4)\)** \[ (a^2 - b^2)(a^4 + a^2b^2 + b^4) \] To multiply these two polynomials, we distribute each term in the first polynomial to each term in the second polynomial: \[ \begin{align*} & = a^2 \cdot a^4 + a^2 \cdot a^2b^2 + a^2 \cdot b^4 \\ & \quad - b^2 \cdot a^4 - b^2 \cdot a^2b^2 - b^2 \cdot b^4 \\ & = a^{6} + a^{4}b^{2} + a^{2}b^{4} - a^{4}b^{2} - a^{2}b^{4} - b^{6} \end{align*} \] Simplifying by combining like terms: \[ a^{6} - b^{6} \] 3. **Third Step: Mistake Identified** \[ a^{6} - b^{8} \] **Error:** The final expression should be \(a^{6} - b^{6}\), not \(a^{6} - b^{8}\). The mistake likely arose from an incorrect multiplication in the previous step. 4. **Correct Final Expression:** \[ (a + b)(a - b)(a^4 + a^2b^2 + b^4) = a^6 - b^6 \] **Summary:** - The expansion correctly leads to \(a^6 - b^6\). - The last term \( -b^8 \) is incorrect; it should be \( -b^6 \). **Additional Note:** Interestingly, the expression \(a^6 - b^6\) can be further factored using the difference of squares and the difference of cubes: \[ a^6 - b^6 = (a^3)^2 - (b^3)^2 = (a^3 - b^3)(a^3 + b^3) \] \[ = (a - b)(a^2 + ab + b^2)(a + b)(a^2 - ab + b^2) \] This shows the complete factorization of \(a^6 - b^6\), which might be useful depending on the context of your problem.